91Ó°ÊÓ

Compute the curl of the vector fields:

$F\left(x,y\right)=-4x^{2}yi+4x{y}^{2}j$.

Find the areas of the given surfaces in Exercises 21–26.

S is the lower branch of the hyperboloid of two sheets $z^2=x^2+y^2+1$that lies below the annulus determined by $1≤r≤2$ in the xy plane.

In Exercises 21–28, evaluate the multivariate line integral of the given function over the specified curve.

$g(x,y,z)=xyz$, with C the curve parameterized by $\mathbf{r}\left(t\right)=\left(\frac{\sqrt{2}}{3}{t}^3,t^2,t\right)\mathrm{for}1≤t≤4.$

Evaluate the multivariate line integral of the given function over the specified curve.

$f(x,y,z)=e^{x^2+y+z^2}$with $C$the circular helix of radius $1$, centered about the z-axis, and parametrized by$r\left(t\right)=(\cos t,\sin t,t)$from height$0$to$\mathrm{Ï€}$.

Find the divergence and curl of the following vector fields.

$F\left(x,y,z\right)=\left(x-y\right)i+\left(y-z\right)j+\left(z-x\right)k$

Compute the curl of the vector fields:

$F\left(x,y\right)=\cos \left(x+y\right)i+\sin \left(x-y\right)j$.

Integrate the given function over the accompanying surface in Exercises 27–34.

$f(x,z)=e^{-(x^2+z^2)}$, where S is the unit disk centered at the point (0, 2, 0)and in the plane y = 2.

Sketch the vector fields in Exercises 25–32.

$\mathbf{F}(x,y)=\mathbf{i}+\mathbf{j}$

$\mathbf{F}(x,y,z)=e^{z}x\sin y\mathbf{i}+e^z\cos y\mathbf{j}+e^x{\tan }^{-1}y\mathbf{k}$, and $s$ is the surface of the region $W$ that lies within the unit sphere and above the plane $z=\frac{\sqrt{2}}{2}$.

${∫}_C\mathbf{F}(x,y,z)·d\mathbf{r}$, where C is the curve in the plane $x-y+z=20$ and that lies above the curves $y=4$and $y=x^2$ in the$xy$-plane, traversed counterclockwise with respect to$\mathbf{n}=⟨1,-1,1âŸ\copyright$, and where $\mathbf{F}(x,y,z)=$ $(2x-3y+4z)\mathbf{i}+(5x+y-z)\mathbf{j}+(x+4y+2z)\mathbf{k}$.